Question : If $\left(x^2 - \frac{1}{x^2}\right) = 4 \sqrt{6}$ and $x>1$, what is the value of $\left(x^3 - \frac{1}{x^3}\right)?$

Option 1: $20 \sqrt{2}$

Option 2: $24 \sqrt{2}$

Option 3: $18 \sqrt{2}$

Option 4: $22 \sqrt{2}$

Question : If $\left(x^2+\frac{1}{x^2}\right)=6$ and $0<x<1$, what is the value of $x^4-\frac{1}{x^4}$?

Option 1: $24\sqrt{2}$

Option 2: $-24\sqrt{2}$

Option 3: $-12\sqrt{10}$

Option 4: $12\sqrt{10}$

Question : If $\left(x+\frac{1}{x}\right)=2 \sqrt{2}$ and $x>1$, what is the value of $\left(x^6-\frac{1}{x^6}\right)$?

Option 1: $140\sqrt{2}$

Option 2: $116\sqrt{2}$

Option 3: $144\sqrt{2}$

Option 4: $128\sqrt{2}$

Question : If $x^{2} + \frac{1}{x^{2}} = 18$ and $x > 0$, what is the value of $x^{3} + \frac{1}{x^{3}}?$

Option 1: $36 \sqrt{5}$

Option 2: $40 \sqrt{5}$

Option 3: $46 \sqrt{5}$

Option 4: $34 \sqrt{5}$

Question : If $\left(x-\frac{1}{x}\right)^2=12$, what is the value of $\left(x^2-\frac{1}{x^2}\right)$, given that $x>0$?

Option 1: $6 \sqrt{2}$

Option 2: $8 \sqrt{3}$

Option 3: $6 \sqrt{3}$

Option 4: $8 \sqrt{2}$